Molecular characteristics are closely related to the kinds of atoms constituting molecules, or electronic states thereof. Elucidating molecular electronic states enables to perform an analysis on stable molecular structures which minimize molecular energies, transition state structures, normal vibrations, and the like by e.g. analytically obtaining derivations by an energy coordinate (so-called energy gradient method). Also, calculating potential energies with respect to a reaction coordinate in molecular reactions enables to obtain reaction systems, generation systems, reaction intermediates, and transition states, as equilibrium positions. Besides, various properties such as vibrational spectrum, electron spectrum, dipole moment, ionization potential, polarizability, and spin density can be obtained. Thus, elucidating molecular electronic states enables to know various molecular characteristics.
There is known a molecular orbital method, as a method for approximately determining molecular electronic states based on quantum mechanics. The molecular electronic states are represented by molecular orbitals. The molecular orbitals are obtained by solving an equation called Hartree-Fock-Roothaan equation (hereinafter, abbreviated as “HFR equation”). The HFR equation is an equation for determining set of spin orbitals which best approximates a system wave function to a basis state in the case where the system wave function is approximated by a single Slater determinant.
Also, the molecular orbital method is substantially classified, depending on the degree of approximation to be utilized in solving the HFR equation into: an empirically molecular orbital method, as represented by Hueckel method or extended Hueckel method; a semiempirically molecular orbital method, in which actual measurement values are used as computation parameters, while neglecting a specific term of a two-electron integral as a sufficiently small value; and an ab initio molecular orbital method for determining molecular electronic states by completely relying on computation based on the first principle without using actual measurement values except for the physical constants. Whereas the empirically molecular orbital method and the semiempirically molecular orbital method lack in reliability because the computation results depend on the approximation methods, parameters or the like, the ab initio molecular orbital method is superior in a point that the method is free from the drawback. There are known application programs of executing the ab initio molecular orbital method e.g. Gaussian 94/98/03 (product of Gaussian. Inc., U.S.A.) and GAMESS (product of NRCC, U.S.A.).
In the case where molecular electronic states are actually calculated by using the ab initio molecular orbital method, the calculation time is drastically increased, as the number N of atoms constituting a molecule is increased. Generally, the calculation time is conceived to be proportional to the third power or fourth power of the number N of atoms. Therefore, whereas it is possible to calculate the molecular electronic states within a reasonable time if the molecule consists of several atoms, it is impossible to calculate the molecular electronic states within a reasonable time if the molecule is a giant molecule consisting of multitudes of atoms such as polymers, which makes it substantially difficult to calculate the molecular electronic states.
In view of the above, some of the inventors of the invention have developed an elongation method for calculating electronic states of aperiodic polymers. The elongation method is a method for successively calculating electronic states of a targeted polymer by calculating the electronic states, with use of below-mentioned localized molecular orbitals (hereinafter, abbreviated as “LMOs”), each time a fragment is added, in place of using canonical molecular orbitals (hereinafter, abbreviated as “CMOs”) basis, by successively adding monomers as additives (fragments) to an oligomer as a starting material (starting cluster) in such a manner as to trace a polymerization reaction of a polymer so that the starting cluster is elongated into the targeted polymer. The molecular orbital method for determining molecular electronic states by the elongation method is e.g. disclosed in the non-patent document 1, the non-patent document 2, the patent document 1, and the like, and will be summarized as follows.
FIG. 8 is a flowchart showing a molecular orbital computing method for determining molecular electronic states by the elongation method according to a background art. FIG. 9 is a diagram schematically showing molecular orbitals in respective steps to describe the molecular orbital computing method for determining molecular electronic states by the elongation method. FIG. 10 is a diagram for describing computations in adding fragments to active LMOs. FIG. 11 is a diagram for describing successive calculations in the elongation method.
Referring to FIGS. 8 through 11, the molecular orbital computing method for determining molecular electronic states by the elongation method includes a step of determining a starting cluster with respect to a targeted polymer whose electronic states are to be calculated, and obtaining CMOs by atomic orbitals (hereinafter, abbreviated as “AOs”) basis of the starting cluster (S101). The initial starting cluster is a part of the targeted polymer consisting of a certain number of atoms, which includes one end of the targeted polymer, and has a length capable of constructing LMOs, and computing the electronic states by a known molecular orbital computing method. The expression “capable of constructing LMOs” means that an interaction by the atom at one end of the starting cluster does not substantially affect the atom at the other end thereof, and that an interaction by the atom at the other end does not substantially affect the atom at the one end. The length of the starting cluster may vary depending on the kinds of atoms constituting the starting cluster, but may normally be from 10 angstroms to 20 angstroms.
Next, the AOs basis for the molecular orbitals (hereinafter, abbreviated as “MOs”) of the starting cluster is transformed into hybridized atomic orbitals (hereinafter, abbreviated as “HAOs”) basis (Step S102). The CMOs by AOs basis obtained in Step S101 are distributed with respect to the entirety of the starting cluster, as schematically shown in FIG. 9(A). However, the transformation allows the CMOs by AOs basis to reside between the respective atoms of the starting cluster, as schematically shown in FIG. 9(B). The transformation can be computed by using the formulas 21 through 25.SaxSax†Ūal=λax1Ūal  (formula 21)where Sax is an overlap integral between atom “a” and atom “x” (x=b, c, d, e), and Sax+ is a transpose matrix of Sax. In the following, similarly to the above, the superscript suffix “+” represents a transpose matrix. Ūal (l=1, 2, 3, 4) is an eigenfunction of SaxSax+, λax2 is an eigenvalue thereof, and b, c, d, e are respective orbitals of sp3 hybridized orbital.Ux=ŪxXx (x=a, b, c, d, . . . )  (formula 22)where Ux is a transformation matrix, and x=a, b, c, d, . . . .
                              ψ          i          ′                =                                            ∑              r                        ⁢                                          (                                                      ∑                    t                                    ⁢                                                            C                      it                                        ⁢                                          U                      rt                                                                      )                            ⁢                              χ                r                                              =                                    ∑              r                        ⁢                                          C                ir                ′                            ⁢                              χ                r                                                                        (                  formula          ⁢                                          ⁢          23                )            where C is a molecular orbital coefficient, with the original atomic orbital as a basis, and C′ is a molecular orbital coefficient, with the hybridized orbital as a basis.
                              ψ          j          ′                =                                            ∑              s                        ⁢                                          (                                                      ∑                    u                                    ⁢                                                            C                      ju                                        ⁢                                          U                      su                                                                      )                            ⁢                              χ                s                                              =                                    ∑              s                        ⁢                                          C                js                ′                            ⁢                              χ                s                                                                        (                  formula          ⁢                                          ⁢          24                )            where, similarly to the above, ψj′ is the j-th molecular orbital by hybridized orbital basis, and χs is the atomic orbital.F′C′=S′C′E  (formula 25)where F′ is expressed by the formula 25-1, S′ is expressed by the formula 25-2, and C′ is expressed by the formula 25-3.F′=U†FU  (formula 25-1)S′=U†SU  (formula 25-2)C′=U†C  (formula 25-3)
Next, LMOs by AOs basis of the starting cluster which have been localized in such a manner that the phase of the orbital is increased at a specific site are created based on CMOs by HAOs basis of the starting cluster (S103). In creation of LMOs by AOs basis, as schematically shown in FIG. 9(C), there are created frozen LMOs φi which have been localized in such a manner that the phase of the orbital is increased on one end (frozen LMO region A or frozen LMO part) of the starting cluster to which a fragment is not added, and created active LMOs φj which have been localized in such a manner that the phase of the orbital is increased on the other end (active LMO region B or active LMO part) of the starting cluster to which a fragment is added. The starting cluster is sorted into the frozen LMO region and the active LMO region because it is conceived that an interaction between the starting cluster and the fragment occurs solely on the other end (reaction end) of the starting cluster to which the fragment is added, and that an interaction at the one end of the starting cluster to which the fragment is not added may be of a substantially negligible degree. The localization process of creating LMOs by AOs basis can be computed by using the formulas 26 through 31.
                              ϕ          i                =                              sin            ⁢                                                  ⁢            θ            ⁢                                                  ⁢                          ψ              i              ′                                +                      cos            ⁢                                                  ⁢            θ            ⁢                                                  ⁢                          ψ              j              ′                                                          (                  formula          ⁢                                          ⁢          26                )                                          ϕ          j                =                                            -              cos                        ⁢                                                  ⁢            θ            ⁢                                                  ⁢                          ψ              i              ′                                +                      sin            ⁢                                                  ⁢            θ            ⁢                                                  ⁢                          ψ              j              ′                                                          (                  formula          ⁢                                          ⁢          27                )                                                                    ϕ              =                                                (                                                            ∑                      r                                              on                        ⁢                                                                                                  ⁢                        A                                                              ⁢                                          +                                              ∑                        r                                                  on                          ⁢                                                                                                          ⁢                          B                                                                                                      )                                ⁢                                  (                                                            sin                      ⁢                                                                                          ⁢                      θ                      ⁢                                                                                          ⁢                                              C                        ir                        ′                                                              +                                          cos                      ⁢                                                                                          ⁢                      θ                      ⁢                                                                                          ⁢                                              C                        jr                        ′                                                                              )                                ⁢                                  χ                  r                                                                                                        =                                                                    ϕ                    i                                    ⁡                                      (                    A                    )                                                  +                                                      ϕ                    i                                    ⁡                                      (                    B                    )                                                                                                          (                  formula          ⁢                                          ⁢          28                )                                                                                    ϕ                j                            =                                                (                                                            ∑                      s                                              on                        ⁢                                                                                                  ⁢                        A                                                              ⁢                                          +                                              ∑                        s                                                  on                          ⁢                                                                                                          ⁢                          B                                                                                                      )                                ⁢                                  (                                                                                    -                        cos                                            ⁢                                                                                          ⁢                      θ                      ⁢                                                                                          ⁢                                              C                        is                        ′                                                              +                                          sin                      ⁢                                                                                          ⁢                      θ                      ⁢                                                                                          ⁢                                              C                        js                        ′                                                                              )                                ⁢                                  χ                  s                                                                                                        =                                                                    ϕ                    j                                    ⁡                                      (                    A                    )                                                  +                                                      ϕ                    j                                    ⁡                                      (                    B                    )                                                                                                          (                  formula          ⁢                                          ⁢          29                )                                          L          ij                =                              〈                                                            ϕ                  i                                ⁡                                  (                  A                  )                                            |                                                ϕ                  i                                ⁡                                  (                  A                  )                                                      〉                    +                      〈                                                            ϕ                  j                                ⁡                                  (                  B                  )                                            |                                                ϕ                  j                                ⁡                                  (                  B                  )                                                      〉                                              (                  formula          ⁢                                          ⁢          30                )                                          L          ij                =                                            α              ij                        ⁢                          sin              2                        ⁢            θ                    +                      2            ⁢                                                  ⁢                          γ              ij                        ⁢            sin            ⁢                                                  ⁢            θ            ⁢                                                  ⁢            cos            ⁢                                                  ⁢            θ                    +                                    β              ij                        ⁢                          cos              2                        ⁢            θ                                              (                  formula          ⁢                                          ⁢          31                )            where αij is expressed by the formula 31-1, βij is expressed by the formula 31-2, γij is expressed by the formula 31-3, θ is expressed by the formula 31-4, and ω is expressed by the formula 31-5.
                              α          ij                =                                            ∑              r                              on                ⁢                                                                  ⁢                A                                      ⁢                                          ∑                s                                  on                  ⁢                                                                          ⁢                  A                                            ⁢                                                C                  ir                  ′                                ⁢                                  C                  is                  ′                                ⁢                                  S                  rs                  ′                                                              +                                    ∑              r                              on                ⁢                                                                  ⁢                B                                      ⁢                                          ∑                s                                  on                  ⁢                                                                          ⁢                  B                                            ⁢                                                C                  jr                  ′                                ⁢                                  C                  js                  ′                                ⁢                                  S                  rs                  ′                                                                                        (                  formula          ⁢                                          ⁢          31          ⁢                      -                    ⁢          1                )                                          β          ij                =                                            ∑              r                              on                ⁢                                                                  ⁢                A                                      ⁢                                          ∑                s                                  on                  ⁢                                                                          ⁢                  A                                            ⁢                                                C                  jr                  ′                                ⁢                                  C                  js                  ′                                ⁢                                  S                  rs                  ′                                                              +                                    ∑              r                              on                ⁢                                                                  ⁢                B                                      ⁢                                          ∑                s                                  on                  ⁢                                                                          ⁢                  B                                            ⁢                                                C                  ir                  ′                                ⁢                                  C                  is                  ′                                ⁢                                  S                  rs                  ′                                                                                        (                  formula          ⁢                                          ⁢          31          ⁢                      -                    ⁢          2                )                                          γ          ij                =                                            ∑              r                              on                ⁢                                                                  ⁢                A                                      ⁢                                          ∑                s                                  on                  ⁢                                                                          ⁢                  A                                            ⁢                                                C                  ir                  ′                                ⁢                                  C                  js                  ′                                ⁢                                  S                  rs                  ′                                                              -                                    ∑                              on                ⁢                                                                  ⁢                B                                      ⁢                                          ∑                                  on                  ⁢                                                                          ⁢                  B                                            ⁢                                                C                  ir                  ′                                ⁢                                  C                  js                  ′                                ⁢                                  S                  rs                  ′                                                                                        (                  formula          ⁢                                          ⁢          31          ⁢                      -                    ⁢          3                )                                          θ          ext                =                  (                                    π              4                        -                          ω              2                                )                                    (                  formula          ⁢                                          ⁢          31          ⁢                      -                    ⁢          4                )                                ω        =                              tan                          -              1                                ⁢                      {                                                            β                  ij                                -                                  α                  ij                                                            2                ⁢                                                                  ⁢                                  γ                  ij                                                      }                                              (                  formula          ⁢                                          ⁢          31          ⁢                      -                    ⁢          5                )            
Next, MO in the case where fragments are added to the starting cluster is computed (S104). The molecular electronic states can be determined by e.g. solving a Fock matrix (F matrix) by a self-consistent field (SCF) method. According to the SCF method, a new electron density is obtained by diagonalizing the F matrix, using an initial electron density. Then, another new electron density is obtained by diagonalizing the F matrix, by using the newly obtained electron density as an initial electron density. This operation is iteratively executed until the electron density defined as the initial electron density is substantially equal to the electron density obtained by diagonalizing the F matrix. The F matrix is solved by conducing the aforementioned procedure in the SCF method.
In the SCF method, normally, diagonalization of the AO-based Fock matrix FAO shown in FIG. 10C is required. With use of LMOs basis, however, as schematically shown in FIGS. 10A and 10B, a fragment (attacking molecule) is exclusively interacted with active LMOs. Accordingly, concerning the Fock matrix FAO, the SCF method is executed solely with respect to the lower right regional parts of FLMO22, FLMO23, FLMO32, and FLMO33, because the respective elements in inverse L-shaped regional parts of FLMO11, FLMO12, FLMO13, FLMO21, and FLMO31 shown in FIG. 10C can be regarded as zero. Thus, as compared with a molecular orbital computing method in which all the systems are processed, the above method is advantageous in reducing the calculation amount, and providing efficient and high-speed computation.
Here, FLMO11 are interaction-related matrix elements in which the respective orbitals of frozen LMOs are acted with the orbitals thereof. FLMO12 are interaction-related matrix elements in which the respective orbitals of frozen LMOs are acted with the orbitals of active LMOs. FLMO13 are interaction-related matrix elements in which the respective orbitals of frozen LMOs are acted with the orbitals of fragments. FLMO21 are interaction-related matrix elements in which the respective orbitals of active LMOs are acted with the orbitals of frozen LMOs. FLMO22 are interaction-related matrix elements in which the respective orbitals of active LMOs are acted with the orbitals thereof. FLMO23 are interaction-related matrix elements in which the respective orbitals of active LMOs are acted with the orbitals of fragments. FLMO31 are interaction-related matrix elements in which the respective orbitals of fragments are acted with the orbitals of frozen LMOs. FLMO32 are interaction-related matrix elements in which the respective orbitals of fragments are acted with the orbitals of active LMOs. FLMO33 are interaction-related matrix elements in which the respective orbitals of fragments are acted with the orbitals thereof.
The fragment-added molecular orbital computation process of computing MO in the case where a fragment is added to the starting cluster is expressed by the formulas 32 and 33. Specifically, computation is implemented solely with respect to the lower right part of the formula 32 partitioned by the broken lines.
where HijOCC (X,Y) is expressed by the formula 33.HijOCC(X,Y)=∫φj(OCC,X)Hφj(OCC,Y)dτ  (formula 33)where φj(OCC,X) is the j-th occupied orbital which has been localized to the region X (X is a frozen region or an active region).
Next, judgment is made as to whether a resultant obtained by adding a fragment to the starting cluster is a targeted polymer (S105). If the judgment result indicates that the resultant is not the targeted polymer, the routine returns to Step S102 by regarding the resultant in Step S104 obtained by adding the fragment to the starting cluster, as a new starting cluster. If, on the other hand, the judgment result indicates that the resultant is the targeted polymer, the molecular orbital computation is ended.
By implementing the aforementioned operations, as shown in FIG. 11A to 11G, the electronic states are successively computed, each time a fragment is added, while successively adding the fragment to the starting cluster. Referring to FIG. 11, the oval-shaped marks represent fragments. For instance, if a targeted material whose electronic states are to be calculated is a polymer, the fragments are monomers.
Since interaction with a fragment is not acted on the end part (frozen AO region) remotely away from the active LMO region, the electronic states can be fixed at the end part. Accordingly, in the iterative calculations from Step S102 through Step S105, the end part can be eliminated from a targeted object to be calculated. Therefore, the iterative calculations from Step S102 through Step S105 are implemented with respect to a region (active AO region) having a certain length. Also, the active AO region is sequentially shifted to the other end of the resultant to which a fragment is added, each time the fragment is added. In this way, the molecular electronic states after the fragment addition can be computed efficiently without lowering calculation precision. The frozen AO region is a region where interaction with a fragment at the frozen LMO is equal to or smaller than a predetermined threshold value (e.g. 10−5 a.u. or 10−6 a.u., a.u. represents atom unit).
In the example schematically shown in FIGS. 11A to 11G, the starting cluster as shown in FIG. 11A consists of two fragments, and five fragments constitute an active AO region, as shown in FIG. 11D. Therefore, as shown in FIG. 1E, if the resultant is constituted of six fragments, a frozen AO region with one fragment is generated. Then, as shown in FIGS. 11F and 11G, each time a fragment is added, the frozen AO region is successively extended toward the other end of the resultant to which the fragment is added. Also, the active AO region is successively shifted toward the other end of the resultant to which the fragment is added.
Here, referring to FIGS. 11A to 11G, the oval-shaped marks shown by the hatched portions represent fragments of an active AO region, and the oval-shaped marks shown by the hollow portions represent fragments of a frozen AO region. Further, in the examples shown in FIGS. 11A to 11G, frozen LMOs and active LMOs are formed respectively in such a manner that the orbitals are localized with respect to two fragments at one end of the resultant to which a fragment is added, and the orbitals are localized with respect to one fragment at the other end of the resultant to which a fragment is added. The region where the frozen LMOs and the active LMOs are formed, in other words, a region (region corresponding to three fragment lengths in FIGS. 11A to 11G) consisting of a frozen LMO region A and an active LMO region B is called as a localized region.
The molecular orbital computing method for determining molecular electronic states by the elongation method according to the background art is a method based on a premise that fragments are sequentially added to a starting cluster with respect to a giant molecule whose electronic states are to be calculated. The method includes: creating LMOs on the starting cluster which have been localized to an active LMO region strongly interacted with the MOs of the fragment by a proper unitary transformation; and solving an eigenvalue problem by the SCF method in association with the CMOs on the fragment to determine the electronic states of the entirety of the giant molecule.
The aforementioned localization process requires a unitary transformation. The unitary transformation includes: arbitrarily selecting two CMOs in pairs from the CMOs; transforming the CMOs in pairs into MO which has been localized to a frozen LMO region and an active LMO region, respectively; and iteratively executing the transformation until convergence is seen with respect to all the pairs.
The convergence is particularly slow in a system where non-localization of orbitals is strong. Therefore, the calculation time is unduly increased in a large basis set.
Also, in the localization process, localization is executed after the CMOs are sorted into the MOs which have been localized to the frozen LMO region and the active LMO region, respectively. The transformation is conducted by selecting two orbitals in pairs individually, which requires an unduly long time for convergence of localization, and consequently may lower precision concerning computation results.    non patent document 1: “A theoretical synthesis of polymers by using uniform localization of molecularorbirals: Proposal of an elongation method” by Akira Imamura, Yuriko Aoki, and Koji Maekawa, J. Chem. Phys., Vol. 95, pp. 5419-5431 (1991)    non patent document 2: “Study contents on project”, [online], internet <http://aoki.cube.Kyushu-u.ac.jp/text/contents/JST_project/JST_content_new.html> [retrieved on Aug. 31, 2004]    patent document 1: Japanese Unexamined Patent Publication No. 2003-012567